| L(s) = 1 | − 1.73·2-s + 0.732·3-s + 0.999·4-s − 1.26·6-s − 0.732·7-s + 1.73·8-s − 2.46·9-s + 1.26·11-s + 0.732·12-s + 4·13-s + 1.26·14-s − 5·16-s + 17-s + 4.26·18-s + 5.46·19-s − 0.535·21-s − 2.19·22-s − 2.19·23-s + 1.26·24-s − 6.92·26-s − 4·27-s − 0.732·28-s + 3.46·29-s + 6.73·31-s + 5.19·32-s + 0.928·33-s − 1.73·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.422·3-s + 0.499·4-s − 0.517·6-s − 0.276·7-s + 0.612·8-s − 0.821·9-s + 0.382·11-s + 0.211·12-s + 1.10·13-s + 0.338·14-s − 1.25·16-s + 0.242·17-s + 1.00·18-s + 1.25·19-s − 0.116·21-s − 0.468·22-s − 0.457·23-s + 0.258·24-s − 1.35·26-s − 0.769·27-s − 0.138·28-s + 0.643·29-s + 1.20·31-s + 0.918·32-s + 0.161·33-s − 0.297·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8107401248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8107401248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08053616229160852210872387872, −9.915611041572223019167629459422, −9.382188473077431894215583537339, −8.421416044887131589811179925581, −7.961813511297414577839190731928, −6.74253352063884930516399980113, −5.65148866906243903393819041999, −4.09315559378755493942349670022, −2.77588952502158478813771462910, −1.06565410959825641294290623053,
1.06565410959825641294290623053, 2.77588952502158478813771462910, 4.09315559378755493942349670022, 5.65148866906243903393819041999, 6.74253352063884930516399980113, 7.961813511297414577839190731928, 8.421416044887131589811179925581, 9.382188473077431894215583537339, 9.915611041572223019167629459422, 11.08053616229160852210872387872
